Off-Diagonal Quasicrystals

• Off-diagonal quasicrystals are quasiperiodic systems where aperiodicity arises from modulated hopping terms rather than on-site potentials.
• They exhibit fractal spectra, sub-ballistic quantum transport, and robust topological edge states, enabling applications in photonics and quantum thermodynamics.
• These systems extend to gravitational and cosmological models, offering insights into modified spacetime metrics and novel dark sector phenomena.

An off-diagonal quasicrystal refers to a class of quasiperiodic systems in which the primary structural aperiodicity is imposed on the kinetic, or hopping, terms of the Hamiltonian rather than on on-site potentials. These systems exhibit distinctive spectral, transport, and topological phenomena, and have been investigated both in condensed matter and in gravitational/cosmological frameworks. The off-diagonal quasicrystal concept encompasses finite and infinite dimensional models, classical and quantum dynamical properties, and finds relevance in modern quasicrystal photonics, thermodynamics, and gravitational theory.

1. Formal Definition and Model Constructions

Off-diagonal quasicrystal models are realized by modulating the hopping amplitudes of a tight-binding Hamiltonian in a non-periodic, deterministic (often aperiodic) manner. A canonical example is the one-dimensional off-diagonal Harper or Aubry-André Hamiltonian: $$H = \sum_{n}\left[t_{n}\,c_{n}^{\dagger}c_{n+1} + \mathrm{h.c.}\right]$$ with hopping modulations of the form

$t_n = t_0\left[1+\lambda d_n\right]$

where $d_n$ is a quasiperiodic function, for instance $\cos(2\pi b n + \phi)$ with irrational $b$, or a two-level Fibonacci sequence. The defining property is that the nontrivial aperiodicity resides in the off-diagonal terms, with or without an additional diagonal (potential) modulation (Verbin et al., 2014, Marin, 2011, Zeng et al., 2021, Suo et al., 25 Jan 2026).

In the specific case of the off-diagonal Fibonacci Hamiltonian, the sequence $d_n$ implements the canonical binary order of a Fibonacci quasicrystal via a modulation that takes the values $\pm 1$ according to Fibonacci substitution rules (Verbin et al., 2014, Marin, 2011).

Models exhibiting both diagonal and off-diagonal quasiperiodic modulations, simultaneously, can be written as

$$H = \sum_{n} \left[ t_{n} c_{n}^{\dagger} c_{n+1} + \mathrm{h.c.} \right] + \sum_{n} V_{n} c_{n}^{\dagger} c_{n}$$

with \begin{align*} t_{n} & = t + t_{1}\cos(2\pi\alpha n) \ V_{n} & = V\cos(2\pi\alpha n) \end{align*} where $\alpha$ is irrational (Suo et al., 25 Jan 2026).

In cosmology and gravitational theory, "off-diagonal quasicrystal" refers to metrics of spacetime in which the off-diagonal components (determined by a so-called nonlinear connection or N-connection) acquire quasicrystal-type aperiodic order through generating functions obeying quasicrystal pattern formation equations (Vacaru, 2018, Vacaru et al., 2019, Aschheim et al., 2016).

2. Spectral and Transport Properties

The spectral structure of off-diagonal quasicrystal Hamiltonians is fractal and typically supports singular-continuous measures. For the off-diagonal Fibonacci chain, the dynamical exponents describing quantum transport are strictly sub-ballistic for strong modulation, in contrast to ballistic propagation in periodic systems and zero transport in Anderson-localized systems. Rigorous upper bounds for the wave-packet spreading exponent (αᵤ) can be established via trace map and transfer matrix techniques (Marin, 2011).

The spectrum is a zero-Lebesgue-measure Cantor set, with explicit lower bounds on fractal (box-counting) dimension set by properties of the underlying substitution sequence and modulation strength. The growth of transfer-matrix traces and spectral dimensions are intimately linked, leading to a unified picture of anomalous quantum transport and quasicrystal spectral geometry (Marin, 2011).

Localization properties in models with both diagonal and off-diagonal aperiodicities depend on the relative strength of modulations. A phase transition curve separates the extended ($D \sim 1$), critical ($0 < D < 1$), and localized ($D \sim 0$) phases, with the fractal dimension determined numerically via the mean inverse participation ratio (MIPR) (Suo et al., 25 Jan 2026).

3. Topological Features and Edge States

Off-diagonal quasicrystals can exhibit nontrivial topological phases in one and higher dimensions. In commensurate (periodic) limits, quantized Berry (Zak) phases and associated winding numbers classify the topological phases, predicting the presence of zero- or nonzero-energy edge modes under open boundary conditions.

Incommensurate or aperiodic cases preserve topological equivalence as long as spectral gaps remain open, protected by Chern numbers that are invariant under continuous deformation from the Harper model to the Fibonacci quasicrystal (Verbin et al., 2014). The off-diagonal Harper-Fibonacci family thus underpins the experimental realization of topological pumping, where Chern number quantization manifests as quantized transport across the system upon adiabatic variation of a pump parameter (Verbin et al., 2014).

In higher dimensions, stacking and coupling of off-diagonal quasiperiodic chains can yield Chern insulator phases characterized by nonzero Chern numbers per band and associated chiral edge states (Zeng et al., 2021).

4. Anderson Localization and Criticality

Localization in off-diagonal quasicrystals can be strictly attributed to the quasiperiodic modulation of the hopping amplitudes, in contrast to conventional Anderson localization from random disorder.

For uniform off-diagonal Aubry-André-Harper (AAH) models, full localization does not occur unless mosaic (modulated) structures are introduced. In such mosaic lattices, the system can display Anderson localization purely from deterministic, quasiperiodic off-diagonal terms. The transition between extended, critical, and localized phases can be quantitatively mapped out by the behavior of the fractal dimension and eigenstate spatial structure (Zeng et al., 2021, Suo et al., 25 Jan 2026).

Localized eigenstates in the strongly modulated regime are pinned to site pairs connected by the strongest modulated hoppings, a feature absent from strictly diagonal models (Zeng et al., 2021).

5. Quantum Thermodynamics and Phase Space Diagnostics

Wigner distributions and the associated Wigner entropy allow diagnostic distinction among extended, localized, and critical eigenstates of the off-diagonal quasicrystal in phase space. Extended states show Wigner distributions extended in position and narrowly peaked in momentum, localized states are peaked in position, and critical states are spread in both. The mean Wigner entropy satisfies a well-defined ordering: $$S_W^{(\text{critical})} > S_W^{(\text{extended})} > S_W^{(\text{localized})}$$ (Suo et al., 25 Jan 2026).

As thermodynamic working media, off-diagonal quasicrystals support a diversity of operational modes (heat engine, refrigerator, heater, accelerator) in quantum Otto cycles. The domain of each operational mode depends on localization properties, with the critical regime offering unique thermodynamic functionality (Suo et al., 25 Jan 2026).

6. Off-Diagonal Quasicrystals in Gravity and Cosmology

The concept of off-diagonal quasicrystal generalizes to spacetime geometry via "off-diagonal quasicrystal-like spacetime structures," implemented in modified gravity theories using nonholonomic frames and N-connections. The key geometric feature is the presence of aperiodically ordered off-diagonal metric components constructed from generating functions that solve phase field crystal or related pattern formation PDEs.

Such solutions, termed space–time quasicrystals (STQC), can encode pattern memory effects into the effective Friedmann-Lemaître-Robertson-Walker (FLRW) metric, leading to modified Hubble expansion rates and corrections to inflationary and dark energy phenomenology. Topological properties, nontrivial thermodynamic (Perelman-type) entropies, and distinct torsion-induced dark sector candidates emerge naturally within this geometric framework (Vacaru, 2018, Vacaru et al., 2019, Aschheim et al., 2016).

In the AFDM (Anholonomic Frame Deformation Method) approach, inflation, dark energy, and dark matter phenomena are modeled by the off-diagonal QC order embedded in the spacetime metric and associated connections, with specific PDEs ensuring the formation and stability of quasiperiodic patterns at both classical and quantum cosmological scales.

7. Experimental Realizations and Applications

Off-diagonal quasicrystal structures have been realized in photonic waveguide arrays, where precise spatial placement of waveguides amplifies the control over evanescent couplings, implementing the desired quasiperiodic off-diagonal modulations. This platform enabled direct measurement of topological edge states and quantized pumping across a Fibonacci quasicrystal, confirming key theoretical predictions regarding gap Chern numbers and robust topological transport (Verbin et al., 2014).

In quantum thermodynamics, off-diagonal quasicrystals provide an experimentally accessible medium for exploring phase-dependent work extraction, refrigeration, and Wigner entropy diagnostics, extending the operational scope of quasiperiodic systems beyond traditional solid-state phenomena (Suo et al., 25 Jan 2026).

Table: Key Classes and Features of Off-Diagonal Quasicrystals

Model Class	Defining Feature	Key Phenomena
1D Off-Diagonal Harper	Cosine-modulated quasiperiodic hoppings	Topological Chern phases, edge states
Off-diagonal Fibonacci	Binary substitution modulated hoppings	Fractal spectra, sub-ballistic quantum spread
Off-diagonal mosaic/AAH	Mosaic, site-selective modulations	Anderson localization, mobility edges
Diagonal + Off-diagonal	Simultaneous aperiodicity in both terms	Extended-critical-localized phase diagram
Off-diagonal spacetime QC	Quasicrystalline N-connection in gravity	Modified inflation, DE/DM from QC structure

These structural, spectral, and topological features establish off-diagonal quasicrystals as a rich arena for studying aperiodic order across condensed matter, optical, thermodynamic, and cosmological contexts, with deep connections between deterministic modulations, anomalous transport, geometric flows, and emergent dark sector phenomenology (Verbin et al., 2014, Marin, 2011, Zeng et al., 2021, Suo et al., 25 Jan 2026, Vacaru, 2018, Aschheim et al., 2016, Vacaru et al., 2019).